PHYSICAL REVIEW B 84, 075210 (2011)
Relationship between energetic disorder and open-circuit voltage in bulk
heterojunction organic solar cells
James C. Blakesley* and Dieter Neher
Department of Physics and Astronomy, University of Potsdam, Potsdam, Germany
(Received 8 April 2011; revised manuscript received 13 May 2011; published 12 August 2011)
We simulate organic bulk heterojunction solar cells. The effects of energetic disorder are incorporated through
a Gaussian or exponential model of density of states. Analytical models of open-circuit voltage (VOC ) are derived
from the splitting of quasi-Fermi potentials. Their predictions are backed up by more complex numerical device
simulations including effects such as carrier-density–dependent charge-carrier mobilities. It is predicted that the
VOC depends on: (1) the donor-acceptor energy gap; (2) charge-carrier recombination rates; (3) illumination
intensity; (4) the contact work functions (if not in the pinning regime); and (5) the amount of energetic disorder.
A large degree of energetic disorder, or a high density of traps, is found to cause significant reductions in VOC .
This can explain why VOC is often less than expected in real devices. Energetic disorder also explains the nonideal
temperature and intensity dependence of VOC and the superbimolecular recombination rates observed in many
real bulk heterojunction solar cells.
DOI: 10.1103/PhysRevB.84.075210
PACS number(s): 88.40.jr
I. INTRODUCTION
With internal quantum efficiencies approaching 100%,1
there are two approaches to further enhancing the power
efficiency of bulk heterojunction organic photovoltaics (BHJ
OPVs). The first is to improve the matching between the
absorption of the solar cell and the solar spectrum by
using narrow-band-gap donor materials, alternative acceptor
components, or by building tandem solar cells. The second
approach is to increase the open-circuit voltage (VOC ) without
loss of photocurrent. Unfortunately, our understanding of the
way in which VOC is determined is quite incomplete, and it
is not clear how devices can be engineered to increase VOC
without a loss of quantum efficiency.
Many studies have shown that VOC depends on the energy
gap between the highest occupied molecular orbital (HOMO)
level of the donor component and the lowest unoccupied
molecular orbital (LUMO) level of the acceptor component
of the BHJ.2–4 Perhaps the most famous of these is the work
of Scharber et al.,2 which studied VOC across a wide range of
materials combinations. They found an empirical relationship:
eVOC = Eg − 0.3 eV , where Eg is the donor-acceptor energy
gap, and e is the elementary charge. While this study was
invaluable to engineering a new generation of donor polymers,
it gives little physical insight as to the origins of VOC . The
question has often been asked: What causes the 0.3 eV loss,
and how can it be reduced?
As VOC denotes the voltage at which the total current is
zero, it depends fundamentally on the balance between chargecarrier generation and recombination in the active region.
Recent studies have found that a reliable forecast of VOC can
be achieved by measuring charge-carrier recombination rates
under open-circuit conditions.5,6 They have also been able
to account for the dependence of VOC on light intensity by
considering the variation in carrier density with voltage.
Intriguingly, several studies7–9 have found strong correlations between the energy of emissive charge-transfer (CT)
states and VOC . This observation raises the question of whether
its dependence on the donor-acceptor energy gap is causal, or
just coincidental. Using the principles of detailed balance, it
1098-0121/2011/84(7)/075210(12)
was argued that the CT state energy would determine VOC
if radiative CT state decay is the limiting loss mechanism.10
It has been suggested that reducing CT state emission might
be a route to improving device performance, though many
uncertainties remain, such as the effects of nonradiative
losses.11
In recent years, it has become clear that the energetic
structure of solid films of conjugated materials is complex,
and the reduction of a model to simple, well-defined, energy
levels is an oversimplification. For example, inhomogeneous
broadening of HOMO or LUMO energy levels (energetic
disorder) is typically thought to lead to a spread of states
across an energy range of 0.1 eV or more. When charge
carriers are photogenerated, they will inevitably relax within
this distribution of states, leading to a loss of energy. In
devices, this could be reflected in a drop in VOC . A couple of
recent publications have predicted such behavior in a Gaussian
density of states (DOS).12,13 In this paper, we expand on this
work using both Gaussian and exponential DOS distributions
with analytical and numerical models. In the first section,
an analytical model for VOC in a system with energetic
disorder is derived. Next, we introduce a numerical device
simulation including realistic physical phenomena, such as
contact mirror-charge potentials, field-assisted dissociation,
and density-dependent charge-carrier mobilities. Finally, we
compare the results of the numerical simulations to the
predictions of the analytical model.
II. ANALYTICAL MODEL
In a BHJ solar cell, the rate of recombination of charge
carriers is dependent on the carrier density. A higher carrier
density results in a higher recombination rate. In turn, the
carrier density of a device depends on the voltage. As voltage is
increased into forward bias, the internal field due to the built-in
voltage is reduced, and the carrier density increases. In the
most general approach to predicting VOC , VOC is the voltage
at which the recombination rate exactly matches the photogeneration rate. This was demonstrated neatly by Maurano
075210-1
©2011 American Physical Society
JAMES C. BLAKESLEY AND DIETER NEHER
Evac
ev(x)
(
PHYSICAL REVIEW B 84, 075210 (2011)
−
device is at VOC , then the local recombination rate, R(x), is
equal to the local generation rate:
)
R (x) = G,
ELUMO,A
E’Fe(x)
electron QFP
where x is the depth within the active region. At VOC , no net
current flows by definition. It follows that electron and hole
currents must both be zero, and the quasi-Fermi potentials
(QFPs) for electrons and holes will therefore be constant and
aligned with the Fermi levels of the electron- and hole-injecting
contacts, respectively. The electron and hole QFPs will be
separated by eVOC . We can then write the hole and electron
quasi-Fermi potentials relative to their respective energy levels
as (see Fig. 1):
Δ
metal
eVOC
hole QFP
metal
Δ
(1)
E’Fh(x)
EHOMO,D
L
EF h (x) = φh + ev (x)
(2)
EF e (x) = φe − ev (x) − eVOC + eVBI ,
(3)
and
FIG. 1. (Color online) Energy-level diagram showing an OPV
device at open-circuit voltage. Evac is the vacuum level, and all other
symbols are as defined in the text. All energy intervals are defined as
positive as shown in this diagram.
et al.,6 who measured recombination rates and carrier densities
under open-circuit conditions at different light intensities.
Recombination rates were parameterized by an empirical
power-law dependence on carrier density, while carrier density
was parameterized by an empirical exponential dependence on
voltage. When combined, these gave a logarithmic dependence
of VOC upon light intensity, in agreement with experiments.
While this general approach is able to describe the behavior
of many different material systems, it is thoroughly empirical
and gives little physical insight into the origins of differences
between materials. In contrast, the ideal BHJ diode model
derived by Koster14 represents a specific case of this general
behavior. It predicts VOC in terms of physical parameters,
such as the donor-acceptor energy gap and recombination rate
coefficient. While insightful, this model fails to predict the
precise dependence of VOC on light intensity for the majority
of real devices. In this section, we begin by deriving once again
VOC for an ideal device. We then demonstrate that, by including
the effects of energetic disorder, we are able to reproduce the
more general, nonideal, behavior observed in real devices.
Note that VOC in real devices is also affected by external
device properties, such as series and shunt resistances. These
can particularly complicate the accurate measurement of true
cell VOC . In this paper, we focus on the fundamental physical
limits of VOC , and such external issues are ignored.
We start by deriving VOC for a system without disorder by
considering Fermi-level splitting. Consider the energy-level
diagram in Fig. 1, which shows a BHJ device at VOC . The
device consists of an active layer sandwiched between two
metallic electrodes. The active layer represents the donoracceptor blend. It is treated as an effective medium in which
the electron transport level is defined by the LUMO of the
acceptor material, while the hole transport level is defined by
the HOMO of the donor material.
We apply a constant electron-hole pair generation rate,
G, throughout the active region. At VOC , the generation of
charge carriers must be exactly balanced by charge-carrier
recombination; otherwise a net current would result. If all
carriers recombine close to where they are generated when the
where v(x) is the local internal electrical potential due to
external voltage, built-in voltage, VBI , and space-charge, and
φh and φe are the hole and electron injection barriers,
respectively. VBI is defined here as the difference in the
contact work functions. For the moment, we treat the hole
and electron energy levels as narrow bands and apply the
Boltzmann approximation to obtain the respective local carrier
densities for holes and electrons:
−φh − ev (x)
nh (x) = Nh exp
(4)
kB T
and
−φe + ev (x) + eVOC − eVBI
ne (x) = Ne exp
kB T
(5)
where Nh and Ne are the total density of states for holes and
electrons (in the donor-acceptor blend), respectively, kB is
Boltzmann’s constant, and T is temperature. The rate of chargecarrier recombination is critical to the determination of VOC ,
though the physics underlying it are not fully understood. One
fact that is well established is that recombination rates depend
on the densities of both charge carriers. To begin with, we
assume that direct bimolecular recombination is the dominant
form of recombination. In this case, the recombination rate can
be written as:
R (x) = γ nh (x) ne (x) ,
(6)
where γ is the recombination rate coefficient. The Langevin
formula,15 γL = e (μh + μe ) /εε0 , where μh and μe are
the electron and hole mobility, provides an upper limit of
recombination coefficient, giving the rate at which electrons
and holes encounter one another. While the Langevin rate
assumes diffusion in a homogeneous medium, the rate has
been confirmed as reasonable in fine-scale donor-acceptor
blends by Monte Carlo simulations16 as long as electron and
hole mobilities are reasonably balanced. These simulations
assumed that carriers have a 100% chance of recombination
when they encounter each other. In the case that the decay rate
of the Coulombically bound electron-hole pair is low enough
that carriers can diffuse apart from each other again before
recombining, the recombination rate might be significantly
lower than that predicted by Langevin. Indeed, studies using a
075210-2
RELATIONSHIP BETWEEN ENERGETIC DISORDER AND . . .
variety of different techniques have found that recombination
rates are often far below this limit.5,17
We can combine Eqs. (1), (4), (5), and (6) to get a familiar
expression for the open-circuit voltage:14
G
,
(7)
eVOC = Eg + kB T Ln
γ Nh Ne
where Eg (= φh + φe + eVBI = EHOMO,D − ELUMO,A ) is
the effective energy gap for the donor-acceptor blend. This
formula predicts many of the features of previous experimental
observations, such as the direct dependence of VOC on
donor-acceptor energy gap2–4 and recombination rate.5,6 Using
typical values for generation and recombination rates, one
finds that the second term above is in the region of −0.2 to
−0.5 eV at solar intensity. According to this model, the 0.3 eV
loss in the Scharber2 scheme can be explained by the balance
between recombination and generation, while the large scatter
in VOC for different materials can be explained by materialdependent variations in recombination rates. However, as will
be discussed later, this is not the complete story. Most real
devices do not follow the intensity dependence predicted by
the ideal model, and variation in the densities of states should
also be considered.
According to the principles of thermodynamics, the charge
pair generation rate, G, must be the sum of an optical
component, Gopt , which is proportional to the incident photon
flux, and a thermal-generation component, Gtherm . By detailed
balance, the latter must equal the recombination rate when at
thermal equilibrium in the dark, giving:
Eg
Gtherm = γ Nh Ne exp −
.
(8)
kB T
Substituting G = Gopt + Gtherm into Eq. (7), we get:
Gopt
Eg
.
1+
eVOC = Eg + kB T Ln exp −
kB T
Gtherm
(9)
The thermal and optical generation rates can be measured
from the reverse-bias saturation dark current density, J0 =
eLGtherm , and reverse-bias saturation photocurrent density,
Jph = eLGopt , respectively. In the case that recombination at
short circuit is negligible, the latter equates to the short-circuit
current, JSC . We then find
JSC
mkB T
Ln
VOC =
+1 ,
(10)
e
J0
where m = 1 is the ideality factor. This is the same result
as the famous Shockley equation for p-n junction devices,18
despite the fact that here we are modeling BHJ devices. This
relationship between dark current and open-circuit voltage
has been demonstrated experimentally in bilayer OPVs.19,20
Vandewal et al.8 found that good predictions of VOC in BHJs
were made when J0 was derived by detailed balance from
electroluminescence and photocurrent quantum efficiencies
instead of dark current. Experimentally, the ideality factor,
m, is nearly always found to be greater than 1 in OPVs.8,21,22
Previously, we assumed direct bimolecular recombination and
found m = 1 in Eq. (10). If, instead, we write recombination
as a power-law in carrier density, n, in Eq. (6): R ∝ nα ,
PHYSICAL REVIEW B 84, 075210 (2011)
then the ideality factor becomes m = 2/α. In inorganic p-n
junction diodes, m = 2 is found at low voltages, and m =
1 is found at higher voltage.23 The behavior is attributed to
indirect recombination by the trapping of minority carriers,
which is a monomolecular recombination process. OPVs
have heterojunctions instead of p-n junctions, and there
are no minority carriers, though it is conceivable that a
similar recombination process could occur by trapping of
carriers at a heterojunction interface. However, experimental
studies of recombination in BHJ OPVs consistently show that
recombination is bimolecular, or even super-bimolecular at
higher intensities.24–26 This is inconsistent with an ideality
factor greater than 1 according to the previous analysis. Later
herein, we suggest that energetic disorder can explain this
discrepancy.
The electronic states of molecules in an organic semiconductor film are subject to random energetic variations as a
result of factors such as variations in conjugation length,
rotations and kinking of polymer chains, interactions with
neighboring conjugated molecules, impurities and dipoles
from residual solvent molecules, etc. Note that this disorder is
not intrinsic to the molecular structure, but it is a property of
the solid film. It is strongly dependent on film preparation. The
result of this energetic disorder is that the bands in the DOS
corresponding to the HOMO and LUMO states do not have
well-defined onsets. Instead, a tail of low-energy states extends
far into the nominal energy gap (see Fig. 2). Experimental
evidence for such tail states is strong. Several recent experiments have measured their energetic distribution sensitively
in various materials.24,27–30 They found that tails could be
approximated to Gaussian or exponential distributions with
typical widths of the order of 0.1 to 0.2 eV. Other studies
have used energetic disorder to explain various phenomena in
organic semiconductors, such as temperature-, field-, and carrier density–dependent charge-carrier mobility,31–34 reduced
charge injection barriers,35 and increased band bending at
interfaces.36,37 Intrinsic dopant levels are typically very low
in organic semiconductors, such that the tail states are not
completely filled in OLED and OPV operation. The relaxation
of carriers into deep tail states then must be considered as an
important process in determining device performance.
We now derive the expression for VOC Eq. (7) for devices
with energetic disorder. To simulate the effect of a broadly
distributed DOS onset, we use two different model DOS
distributions: (a) Gaussian distribution and (b) exponential
distribution. We write the model DOS as:
Nh/e
(±E ∓ EHOMO,D/LUMO,A )2
,
gh/e (E) = √ exp −
2σ 2
σ 2π
(11a)
Nt,h/e
E − EHOMO,D/LUMO,A
,
(11b)
gh/e (E) =
exp ±
Et
Et
respectively, where E is energy, EHOMO,D and ELUMO,A are
representative of the donor HOMO energy level and acceptor
LUMO energy level, σ quantifies the energetic disorder in the
Gaussian distribution, and Et is the characteristic energy for
the exponential tail distribution. Care should be taken about
the use of “HOMO” or “LUMO” to describe energy levels.
Their use to refer to the frontier orbitals of a single isolated
075210-3
JAMES C. BLAKESLEY AND DIETER NEHER
(a)
PHYSICAL REVIEW B 84, 075210 (2011)
(b)
conduction
band
LUMO
(c)
(d)
tail states
band gap
valence
band
binding
energy
measured VB onset
HOMO
σ
Et
EHOMO
Nt,h
EHOMO
real DOS
Nh
model
DOS
model
DOS
FIG. 2. (Color online) Schematic density of states (DOS) for (a) classical crystalline inorganic semiconductor and (b) disordered organic
semiconductor showing tail states extending into the band gap. (c) and (d) A Gaussian or exponential DOS respectively (red lines) used
to approximate the tail of the valence states. The widths of the model DOS are σ and Et , and the areas enclosed are Nh and Nt,h ,
respectively.
molecule is clear, but their meaning in the context of a realistic
disordered solid film where localized and delocalized states
may exist together is not well defined. Often, in practice, the
terms are used to refer to onset of valence or conduction states
measured experimentally by techniques such as photoemission
spectroscopy or cyclic voltammetry. These might be better
described as valence band (VB) and conduction band (CB)
onsets. Since these onsets are determined by the energy at
which the states first become visible relative to the background
signal, the measured values are subject to the sensitivity of the
experimental method and instrumental broadening. While a
Gaussian or exponential model might not accurately represent
the entire DOS of a real solid, the majority of occupied states
are within the tail of the distribution, below the measured
onset. In choosing a model DOS, we aim only to describe the
shape of the tail-states distribution. EHOMO,D and ELUMO,A
are mathematical terms. Typically, the measured VB and CB
onsets are offset from EHOMO,D or ELUMO,A in the direction
of the energy gap by up to 3 times σ or Et [see Figs. 2(c) and
2(d)].29,36,37
The carrier density is calculated by integrating the FermiDirac distribution across the DOS. For the Gaussian distribution at low38 carrier densities, this can be approximated by a
Boltzmann distribution13
−EF h (x)
σ2
exp
. (12a)
nh (x) = Nh exp
kB T
2 (kB T )2
therefore a lower VOC . This can be understood intuitively
as a loss of energy as charge carriers relax within the
DOS until they occupy low-energy sites, thus bringing the
average energy of positive and negative charge carriers closer
together.
For the exponential distribution, if Et kB T , then the
carrier density for holes can be approximated by:
for holes, where EFh
is the difference between the hole
quasi-Fermi potential and EHOMO,D , and the same applies for
electrons. Substituting in place of Eqs. (4) and (5), we arrive
at a similar result for the open-circuit voltage:
G
eff
,
(7a)
eVOC = Eg + kB T Ln
γ Nh Ne
III. NUMERICAL MODEL
where Eg is replaced by the new effective donor-acceptor
energy gap, Egeff = EHOMO,D − ELUMO,A − σ 2 /kB T . We find
that a broader DOS causes a lower effective energy gap, and
nh (x) = Nt,h exp
−EF h (x)
.
Et
(12b)
and the equivalent for electrons. Now the VOC becomes:
eVOC
G
= Eg + mkB T Ln
γ Nt,h Nt,e
,
(7b)
where Eg = EHOMO,D − ELUMO,A , and m = Et /kB T . The
kB T in Eq. (7) is replaced by mkB T , which again leads to
loss of VOC . This substitution can also be made for Eqs. (8),
(9), and (10). Equation (7a) predicts ideal diode behavior for
a Gaussian DOS, while Eq. (7b) predicts nonideal behavior
for an exponential DOS. In both cases, eVOC maintains
its one-to-one dependence on the donor-acceptor energy
gap, though the tail states are responsible for an additional
loss.
In deriving an analytical model for VOC , it is necessary
to simplify many aspects of the device physics. Using a
drift-diffusion device model that incorporates a Gaussian
or exponential density of states, we are able to test the
robustness of the results in more realistic physical conditions.
In our model, we include the effects of surface recombination
at the electrodes, field-dependent geminate-pair separation,
and carrier-density–dependent charge-carrier mobilities. To
simplify matters, we assume that acceptor electron and donor
075210-4
RELATIONSHIP BETWEEN ENERGETIC DISORDER AND . . .
hole mobilities, densities of states, and disorder are identical
throughout.
The model uses the standard drift-diffusion form to calculate the current density for both holes and electrons:
J = −eμn
dv
dn
± eD ,
dx
dx
(14)
where ε0 is the permittivity of free space, and ε is the
relative permittivity, under the boundary condition that the
voltage dropped across the active region is equal to the
applied voltage minus the difference in electrode work
functions.
There is currently a debate as to whether free-carrier
generation in BHJ OPVs is electric field dependent or not.
It certainly seems to depend on the material in question. There
is clear evidence of field-dependent carrier generation in some
single-component OPVs and all-polymer BHJ devices.39,40
To take account of this, we used the approach of Koster
et al.41 It is assumed that all photogenerated electron-hole
pairs are generated in an initially Coulombically bound state
(polaron pairs or geminate pairs). These bound pairs can either
dissociate into free carriers with probability p(x), or decay
to the ground state. Similarly, bimolecular recombination
involves the coming together of free electrons and holes to
form bound pairs at rate of R(x). A fraction, p(x), of these will
dissociate back into free carriers again without decaying to the
ground state. The probability of dissociation depends on the
binding energy and decay rate of the bound state, temperature,
electric field, the free-carrier mobility, and the structure of
the donor/acceptor interface in a BHJ. To simplify this, we
use the approach of Braun.42 In this model, a field-dependent
dissociation rate competes with a field-independent decay rate.
Here, we write the explicit field and mobility dependence of
dissociation, while the all other parameters are summarized by
a single material-dependent constant, β (see note [43] for an
explanation of β):
p(x) =
β(μh + μe )
√
√
β(μh + μe ) + −2b/J1 (2 −2b)
Finally, the current continuity equation under steady-state
conditions is applied for electrons and holes:41
dJ
+ [1 − p (x)] R (x) − p (x) G = 0
dx
(16)
(13)
where J is current density, n is carrier density, μ is mobility,
and D is the diffusion coefficient. This is solved simultaneously
with the one-dimensional Poisson’s equation:
d 2v
e (ne − nh )
=
,
dx 2
εε0
PHYSICAL REVIEW B 84, 075210 (2011)
(15)
where J1 is the first-order Bessel function, b =
e3 F (x) /8π εε0 kB2 T 2 , and F(x) is the electric field. This is
only a rough approximation of true dissociation—simulations
have shown that disorder complicates matters44 and that a
broad distribution of binding energies should be considered.45
We use direct bimolecular recombination rates according to
Langevin to estimate the rate at which electrons encounter
holes, R (x) = γL ne nh ,15 with γL = e (μh + μe ) /εε0 . Note
that an increase in mobility reduces the probability of geminate
recombination without disturbing the equilibrium between free
carriers and bound carriers, since carriers will also encounter
each other at a faster rate.
To include the effects of bound-pair dissociation in the
analytical model presented earlier, we note that the equilibrium
condition is now pG = (1 − p) R.14 Making the assumption
that the field is close to zero in open-circuit conditions,
we find that Eqs. (7), (7a), (7b), and (7c) (below) can be
approximated by using a reduced recombination coefficient,
γ = γL /β (μh + μe ). Studies of bimolecular recombination
rates in blends of poly(3-hexyl-thiophene) (P3HT) with a
fullerene derivative have revealed values of the order of
10−1 to 10−3 of the Langevin rate.5,17,26 Similar factors have
been observed in some other materials,46 though it is not
yet clear whether this behavior is general to all efficient
BHJ materials. These reduced recombination rates could be
caused by charge carriers coming into contact with each other
but not recombining. Such materials also happen to be ones
in which field-dependent dissociation is not an important
mechanism,47,48 indicating that the bound-pair decay rate is
low, that the bound-state binding energy is small, or that
free-carrier mobilities are high, and hence bound pairs are most
likely dissociated again before they have a chance to decay
to the ground state.11 In these cases, values for β (μh + μe )
of order 100 are appropriate to model the suppression of
Langevin recombination. For comparison, simulations were
also performed using the conventional non-field-dependent
continuity equation: dJ /dx + R − G = 0, using Langevin
recombination rates. To include the effects of energetic disorder, we need to consider two factors. Firstly, the Boltzmann
approximation does not always apply for the carrier density
as a function of Fermi potential. The diffusion constant, D,
in Eq. (13) must be calculated as a function of carrier density
according to the generalized Einstein relation.49 Secondly, in
a system with energetic disorder, it has been established33 that
mobility increases with charge-carrier density as a result of
the filling up of the lowest energy states. The exact manner in
which the mobility varies depends on the model used. Here, we
compare two models: the Gaussian disorder model (GDM)31
and the mobility edge (ME) (also called multiple trapping and
retrapping) model.50 Both models have been successfully used
to model charge transport in disordered materials,32,34,51,52
and both predict an increase in charge-carrier mobility with
increasing charge-carrier densities. The former is applied to a
Gaussian density of states, as in Eq. (11a), and assumes that all
sites are highly localized, with charge transport occurring by
hopping between these localized states. The carrier-density
dependence of mobility for the GDM has been calculated
using various methods.53–55 Here, we use a variant of the
parameterization scheme of Pasveer53 to calculate the mobility
for each carrier type as a function of x:
1
μ (x) = μ0 exp −0.42σ̂ 2 + (σ̂ 2 − σ̂ ) [2n (x) /N]δ
2
075210-5
ln(σ̂ 2 − σ̂ ) − 0.3266
δ=2
,
σ̂ 2
(17a)
JAMES C. BLAKESLEY AND DIETER NEHER
acceptor
conduction band
PHYSICAL REVIEW B 84, 075210 (2011)
density of states
ELUMO,A
trap states
(t-t)
(f-t)
(f-f)
EHOMO,D
acceptor
donor
valence band
energy
donor
FIG. 3. (Color online) Schematic diagram illustrating different
recombination modes in mobility-edge model. (t-t) trapped-carrier
to trapped-carrier recombination; (f-t) free-carrier to trapped-carrier
recombination; and (f-f) free-carrier to free-carrier recombination.
where μ0 is the high-temperature limit of mobility, σ̂ =
σ/kB T is the amount of disorder relative to thermal energy,
and N is the total density of states.
For the mobility edge model, it is assumed that there exists
a dense band of delocalized states with effective density N at
energy EHOMO,D or ELUMO,A , through which charge transport
occurs with mobility μ0 . An exponential tail of trap states
extends from the band into the band gap with distribution as
given in Eq. (11b). Assuming that no charge transport can
occur between trap states, the effective mobility is given by:50
nf
μ (x) = μ0
,
(17b)
nf + nt
where nf is the density of free carriers in the delocalized
band, and nt is the density of carriers in the trap states. Under
the assumption that trapping and detrapping are sufficiently
rapid, thermal equilibrium is established between these two
populations. They are determined by the relative position of
the quasi-Fermi potential, EF :
E
nf = N exp − F ,
kB T
E
nt ≈ Nt exp − F ,
(18)
Et
n = nf + nt .
There are three possibilities for modeling recombination in
the mobility-edge model (see Fig. 3): (1) Recombination can
only occur between free electrons and free holes (f-f). In this
case, the trap distribution plays no role in determining VOC , and
Eq. (7) remains unchanged. (2) Bimolecular recombination
can occur between free carriers and trapped charge (f-t).
In BHJ devices, this process differs fundamentally from
Shockley-Read-Hall recombination, since electrons and holes
do not coexist in the same material component. Instead, it
can be considered that free carriers in one component are
attracted to the trapped charge in the other component. If
the trapped charge is sufficiently close to a heterojunction
interface, then the mutual attraction could cause it to detrap
and recombine with the free carrier. In this case, we apply a
bimolecular recombination rate following the Langevin form
R = eμ0 (ne nh,f + nh ne,f )/εε0 . (3) Carriers trapped in the tail
states are allowed to recombine directly with other trapped
carriers (t-t). We model this using a bimolecular recombination
rate, R = 2eμ0 ne nh /εε0 . While the third case is theoretically
interesting, it seems physically unlikely that t-t recombination
would contribute significantly to the recombination current in
a real device. We study both cases 2 and 3 here, though case 2
should be considered the most realistic model.
Notice that nf increases superlinearly with n. Considering
case 2, this leads to an apparent increase in the bimolecular recombination rate at higher carrier densities. Such
a phenomenon might explain the various observations of
recombination rates with power-law density dependencies
greater than 2.24–26 If nt nf , as might be the case under
typical OPV operating conditions, the process is dominated
by recombination between free carriers and trapped carriers.
Assuming hole and electron densities are equal, this gives a
power-law behavior R ∝ nα with α ≈ 1 + (Et /kB T ). In this
situation, we need to revise Eq. (7b) in order to predict VOC ,
giving:
G
,
eVOC = Eg + mkB T Ln
γ Nt N
2
m=
,
(7c)
1 + kB T /Et
γ = eμ0 /εε0 .
For the following numerical simulations, devices of thickness L = 100 nm were simulated by dividing them into
at least 400 equally spaced grid points. For both disorder
models, the following parameters were used: mobility μ0 =
10−3 cm2 /Vs, density of states N = 1021 cm−3 , donor-acceptor
energy gap Eg = 2 eV, relative permittivity ε = 3.5, and a
dissociation parameter β = 5 × 104 Vs/cm2 . All simulations
were performed at T = 300 K. Boundary conditions for the
numerical simulation were that the electron and hole quasiFermi potentials in the first and last grid points were aligned
with the contact Fermi levels, i.e., EF (0) = φh , EF (L) =
φe . Studies of injection currents35,56–58 have demonstrated
that the reduction of injection barriers by a mirror-charge
potential is an important effect when injection barriers are
large. To account for this, an additional term, v (x) was added
to the electron potential and subtracted from the hole potential:
1
1
e
+
.
(19)
v(x) =
16π εε0 x
L−x
When the injection barrier is small, the mirror-charge
potential can be ignored due to screening from the large carrier
density at the contact. However, when the injection barriers are
large, it creates an additional surface recombination current
that can significantly affect open-circuit voltage, as reported
in the results section. The simulation also reproduces well the
high- and low-field injection currents predicted by Emtage and
O’Dwyer56 in the absence of energetic disorder.
075210-6
RELATIONSHIP BETWEEN ENERGETIC DISORDER AND . . .
(b)
increasing disorder
-2
Current density (mA cm )
increasing disorder
0.0
-2
Current density (mA cm )
(a)
PHYSICAL REVIEW B 84, 075210 (2011)
low intensity
= 100 meV
-0.1
=0
0.0
0.5
0
= 100 meV
-10
high intensity
=0
1.0
0.0
0.5
Voltage (V)
(d)
increasing disorder
-2
Current density (mA cm )
increasing disorder
= 125 meV
-0.1
= 100 meV
=0
low intensity
0.0
0.5
0
= 100 meV
-10
=0
high intensity
1.0
0.0
0.5
Voltage (V)
1.0
1.5
Voltage (V)
1.4
1.6
1.4
VOC (V)
1.2
1.0
0.8
full simulation
without field-dep. generation
eq. (7a)
0.6
0
20
40
60
80
Energetic disorder,
/ meV
0
25
50
75
100
125
(f)
(e)
VOC (V)
1.5
Voltage (V)
0.0
-2
Current density (mA cm )
(c)
1.0
1.2
1.0
0.8
0.6
10
100 120 140
18
10
19
10
20
10
21
10
22
10
23
-3 -1
Generation rate, G (cm s )
(meV)
FIG. 4. (Color online) Simulations of BHJ devices with Gaussian disorder. (a) and (b) Simulated current-voltage curves for varying degrees
of disorder (σ = 0, 25, 50, 75, 100, and 125 meV) at low (G = 1020 cm−3 s−1 ) and high (G = 1022 cm−3 s−1 ) intensities, respectively. (Dashed red
lines are dark currents.) (c) and (d) Same as (a) and (b) except without field-dependent generation and recombination rates. (e) VOC vs. disorder at
low intensity. Filled symbols: using field-dependent continuity equation [Eq. (16)]. Open symbols: using field-independent continuity equation.
Solid line: Eq. (7a) with effective recombination coefficient γ = γ L /β(μh + μe ). Dashed line: Eq. (7a) with γ = γ L . (f) VOC vs. intensity with
varying disorder and field-dependent generation and recombination. Symbols: numerical simulation; solid lines: Eq. (7a).
IV. RESULTS
First, we consider devices with Ohmic contacts, where
there is no injection barrier for electrons or holes. Figure 4
shows the results of numerical simulations of devices using the
GDM with varying degrees of disorder. Figures 4(a) and 4(b)
shows simulated current-voltage curves for devices at low and
high intensities, respectively. The latter (G = 1022 cm−3 s−1 )
is comparable to 1 sun intensity in typical OPVs. Without
disorder, a very high fill factor is observed. As disorder is
increased, the fill factor drops dramatically, and the simulation
starts to resemble more realistic devices. The drop in fill factor
is attributable to the drop in mobility with increasing disorder.
This occurs through two mechanisms. Firstly, dissociation
of bound electron-hole pairs becomes less efficient, leading
to large geminate losses at low electric fields. Secondly,
bimolecular recombination losses become more predominant
as charge-carrier extraction becomes slower. This is especially
true at high intensities.59,60 For comparison, Figs. 4(c) and
4(d) shows the same curves for simulations in which the field
dependence of free-carrier generation and recombination was
neglected. The impact of disorder is stronger in Figs. 4(a)
and 4(b) as low disorder results in a reduced recombination
rate, while high disorder results in a strongly field-dependent
generation rate.
075210-7
(a)
0.0
(c)
PHYSICAL REVIEW B 84, 075210 (2011)
(b) 1.4
E t = 100 meV
f-f
increasing trap density
10
1.2
19
VOC(V)
Current density (mA cm-2)
JAMES C. BLAKESLEY AND DIETER NEHER
-0.1
10
18
10
16
0.0
0.8
13
0.5
1.0
Voltage (V)
(d)
VOC(V)
VOC(V)
1.4
Nt(cm) E(meV)
t
20
50
1019
100
1018
10
150
1.2
1.1
eq. (7c)
eq. (7)
1.0
18
19
20
21
22
14
15
16
17
18
19
20
21
10 10 10 10 10 10 10-3 10 10
Trap density, Nt (cm )
1.5
1.3
f-f+
f-t+
t-t
eq. (7)
eq. (7b)
eq. (7c)
17
Nt = 10 cm-3
f-f+
f-t
E t = 100 meV
simulation
1.0
23
24
eq. (7)
eq. (7b)
1.5
1.4
1.3
1.2
1.1
1.0
0.9
-3
Nt(cm ) E (meV)
t
50
10 20
17
100
10
1017
150
18
10
10 10 10 10 10 10 10
Generation rate,G (cm-3s-1)
19
20
21
22
23
10 10 10 10 10
Generation rate,G (cm-3s-1)
FIG. 5. (Color online) Simulations of BHJ devices with exponential density of trap states. (a) Simulated current-voltage curves with Et =
100 meV and varying trap densities (Nt = 1016 , 1017 , 1018 , and 1019 cm−3 ) at low (G = 1020 cm−3 s−1 ) intensity. (b) VOC vs. trap density for the
same conditions. Filled symbols: including trap-to-trap recombination (t-t); open symbols: excluding trapped-to-trapped recombination, but
including free-to-trapped-carrier recombination (f-t). (c) VOC vs. intensity excluding trapped-to-trapped recombination for different trap distributions. (d) VOC vs. intensity including trapped-to-trapped recombination for different trap distributions. Lines: analytical solutions from text.
Concurrent with the decline in fill factor, a large decrease
in VOC is seen as disorder is increased. Figure 4(e) plots VOC
as a function of disorder at low intensity, and Fig. 4(f) shows
the dependence of VOC on intensity for varying disorder. Also
shown are the VOC values predicted from Eq. (7a). This predicts
well the results of the simulations at low intensities. At high
intensities, and large quantities of disorder, the simulated VOC
starts to grow more quickly with intensity than predicted by the
ideal diode equation, and Eq. (7a) no longer applies. Instead, it
is necessary to include an ideality factor to fit the predictions,
despite the fact that recombination is still strictly bimolecular.
The nonideality arises from the failure of the Boltzmann
approximation in Eq. (12a)—the real carrier density varies
more rapidly with the Fermi potential than predicted there.
Figure 5 shows the results of simulations using the ME
model with exponential density of trap states. Simulations
were performed both with and without the inclusion of
recombination of trapped electrons with trapped holes (t-t).
In both cases, recombination was allowed between trapped
electrons and free holes and vice versa (f-t). Figure 5(a)
shows the simulated current-voltage characteristics as the
density of trap states is varied at low intensities without (t-t)
recombination. Once again, both the fill factor and VOC fall as
the density of trap states is increased. The decline in fill factor
is not as dramatic as that noted previously since the free-carrier
mobility remains unchanged in this case, and therefore boundpair dissociation is not affected by the presence of traps in this
model. One should be careful not to pay too much attention to
the simulated fill factors, however, because the effect of disorder and trapping on the field dependence of pair dissociation
is still quite debatable.44,45 Figure 5(b) shows the variation of
VOC with trap density at low intensity, while Figs. 5(c) and
5(d) shows the variation of VOC with intensity for different
trap distributions. When (t-t) recombination is allowed, VOC
is predicted reasonably well by Eq. (7b) at low intensities
and high trap densities. When (t-t) recombination is excluded,
Eq. (7c) predicts VOC . In both cases, in the limit of high
intensity or low trap density, we reach a point at which the traps
become completely filled, and the carriers become blind to the
presence of traps. In this case, VOC is limited by the properties
of band-like transport only, and the behavior is as predicted by
Eq. (7) for ideal diodes without disorder. Therefore, according
to the ME model, there is a critical trap density, below which
VOC is not reduced. For 1 sun intensity and assuming Et =
100 meV, this limit is of the order of 1017 to 1018 cm−3 .
Figure 6 shows energy-level diagrams for devices simulated
under open-circuit conditions with and without Gaussian
disorder. It is interesting to note that the electric field in
the middle of the device is almost the same in both cases,
despite the voltage being reduced by a factor by about 0.8 V
when disorder is included. The additional internal voltage is
almost entirely taken up by an increase in band bending near
to the contact interfaces. Recently, such band bending was
observed in Kelvin-probe studies at various metal/conjugated
075210-8
RELATIONSHIP BETWEEN ENERGETIC DISORDER AND . . .
(a) no disorder
Evac
ELUMO,A
Energy
quasi-Fermi
eVOC
potenals
EHOMO,D
Evac
(b) 150 meV disorder
Energy
eVOC
0
20
40
60
80
100
Depth (nm)
FIG. 6. (Color online) Simulated energy diagrams in open-circuit
conditions with G = 1020 cm−3 s−1 and (a) no disorder. (b) σ =
150 meV Gaussian disorder.
polymer interfaces.37 The magnitude of the band bending was
consistent with predictions of models including a significant
amount of energetic disorder. More disorder causes more band
bending, and it was noted13 that the predicted loss of VOC due
to Gaussian disorder is equal in magnitude to the increase in
band bending. It has been suggested that band bending directly
causes a loss of VOC .4,61,62 However, such a causal link is
difficult to demonstrate, especially since there is no need to
explicitly include band bending in the derivation of Eq. (7a).
At least it is possible to state that increased band bending is
concomitant with a loss of VOC . For this reason, we advocate
further experimental investigations into the link between band
bending and VOC .
PHYSICAL REVIEW B 84, 075210 (2011)
In a simple metal-insulator-metal (MIM) model of a
BHJ device, VOC cannot exceed the built-in voltage of the
contacts. Biasing beyond the built-in voltage will reverse the
internal electric field and therefore also the direction of the
photocurrent. This limit does not apply to bilayer solar cells,63
where reverse photocurrents are blocked.
Experiments on poly(phenylene-vinylene) (PPV) derivatives blended with different acceptor molecules4,64 found that
eVOC had a one-to-one dependence on cathode work function
when the electron injection barrier was large. However, when
the cathode work function was small enough to make an
Ohmic contact to the acceptor component, VOC became
independent of cathode work function, but instead had a
one-to-one dependence on the acceptor reduction potential.
In other words, VOC can be limited either by the effective
donor-acceptor energy gap, or by the difference in contact
work functions. In the analytical model derived earlier, it was
assumed that the contacts were completely selective (each
quasi-Fermi potential was aligned to one contact only), and the
effects of surface recombination currents were ignored. This
leads to the elimination of contact work functions from the
model. In reality, there will be an additional loss mechanism
besides bimolecular recombination as some carriers leave the
device by drift and/or diffusion to the contacts. In opencircuit conditions, these additional “surface” recombination
currents must be equal for electrons and holes at each contact
in order to maintain steady-state charge densities in the
contacts. Therefore, if either polarity of charge carrier is
blocked from reaching a contact, then this loss vanishes. With
Ohmic contacts, a large carrier density is present near the
electrode interface (as verified by our own recent experiments
[37]). This charge causes carriers of the opposite polarity to
recombine before reaching the contact. They are also repelled
by the strong band bending present near the interface. The
agreement between the analytical model and the numerical
simulations confirms that surface recombination losses are
negligible under normal conditions. The concept that surface
recombination is negligible for Ohmic contacts has been
demonstrated theoretically,65 while recent experiments have
shown that nonbimolecular losses are negligible at VOC in
P3HT:methanofullerene solar cells.66 By simulating devices
increasing
injection barrier
-0.1
0.0
0.5
(a)
Simulated VOC (V)
φ= 0
1.0
1.0
0.8
S=0
0.6
bipolar
device
0.4
hole-only
device
S=1
0.2
0.0
S=0
(b)
0.0
0.5
1.0
1.5
2.0
∇
-2
no disorder
100meV
Gaussian disorder
1.2
0.0
∇
Current density (mA cm )
1.4
Electron injection barrier, φ (eV)
Voltage (V)
FIG. 7. (Color online) Simulations of BHJ devices with Gaussian disorder and varying electron injection barriers. (a) Current-voltage
characteristics for devices with σ = 100 meV and G = 1020 cm−3 s−1 . (b) VOC vs. electron injection barrier under the same conditions with and
without disorder.
075210-9
JAMES C. BLAKESLEY AND DIETER NEHER
PHYSICAL REVIEW B 84, 075210 (2011)
of different thicknesses (not shown), we found that VOC begins
to decline significantly as the thickness is reduced below about
20 nm. When the image potential [Eq. (19)] was omitted
from the simulations, VOC remained independent of device
thickness. This indicates that surface recombination driven by
recombination between carriers and their own mirror charge
can become significant in devices with Ohmic contacts only
when the device thickness is of the order of the Coulomb
capture radius (rC = e2 /4π εε0 kB T ) or less.
If a significant injection barrier is present at one of the
contacts, then there is expected to be a significant surface
recombination current as photogenerated electrons and holes
flow toward the contact in an attempt to establish an equilibrium with the contact Fermi level, while the blocking effect
discussed previously is removed. This reduces VOC . Naturally,
a homogeneous device with two identical contacts should have
no VOC at all. This was tested by running simulations of devices
with varying electron injection barriers for the low workfunction contact. The resulting current-voltage characteristics
for devices with 100 meV Gaussian disorder are shown in
Fig. 7(a). For small injection barriers, changes in contact
work function have little effect. When the injection barrier
becomes larger than about 0.4 eV, the whole current-voltage
characteristic shifts to lower voltages, and the VOC is reduced.
Figure 7(b) shows VOC as a function of the electron injection
barrier. Interestingly, a z-curve emerges that is reminiscent of
the Fermi-level pinning behavior observed in photoemission
spectroscopy and Kelvin studies36,37,67 of conjugated materials
deposited on substrates with different work functions. When
the contact Fermi level lies close to either EHOMO,D or
ELUMO,A , VOC is independent of the contact work function.
Between these limits, there is a slope of 1. This is quite
consistent with experimental observations.4,64 As previously
observed, disorder reduces VOC when both contacts are Ohmic.
Notice, that the effect of disorder is exactly halved when only
one of the contacts is pinned. It was shown recently37 that the
limits of Fermi-level pinning in polymer films of more than
about 10 nm thickness were governed by band bending. Again,
these results are suggestive of a strong correlation between
band bending and VOC . However, the simple MIM model in
which the VOC is equal to the built-in voltage minus the amount
of band bending at each contact is not quite right. VOC depends
on intensity, while band bending does not; similarly, band
bending depends on the thickness of the active layer, while
VOC , in principle, does not.
V. DISCUSSION
The core principle of this paper is that energetic disorder
will reduce VOC . Interestingly, this result has also been
predicted for a Guassian DOS using quite different methods.12
It has also long been known that tail states in amorphous
silicon cause a reduction in VOC by trapping minority carriers,
leading to an increase in recombination.68 Together, these
findings demonstrate the robustness of the principle, despite
the different theoretical approaches.
This paper has focused on theoretical aspects of opencircuit voltage. When attempting to apply such theories to
experimental results, we are faced with the problem of not
knowing precisely the DOS of the materials that we are
studying. Take, for example, researchers’ favorite material
system of poly(3-hexyl-thiophene) (P3HT) blended with [6,6]phenyl C61 butyric acid methyl ester (PCBM). The difference
between the onset of oxidation in P3HT and the onset of
reduction in PCBM has been measured at about 0.8 eV,2
while others find that the energy gap between donor and
acceptor HOMO/LUMO onsets is more like 1.1 eV.24 In order
to apply Eq. (7a) to predict VOC , we need to make many
assumptions about the DOS where information is not available.
The energetic disorder in P3HT is quite small, estimated about
σ = 70 meV,69 and we can assume a similar magnitude for
PCBM. As mentioned previously, the appropriate position for
the center of the model DOS could be up to 3σ beyond the
measured onset of valence/conduction states. Therefore, an
appropriate EHOMO,D − ELUMO,A could be ∼1.2 eV, and the
effective energy gap with disorder accounted for will be Egeff ≈
1.0 eV. We have to make another assumption about the total
density of states for electrons and holes. Here, we will assume
1021 cm−3 for each. Transient measurements of recombination
give typical values of the order of γ = 10−12 m3 s−1 ,5,47,70
though these vary greatly depending on material preparation.
Finally, taking G = 1022 cm−3 s−1 for 1 sun intensity, Eq. (7a)
gives us VOC ≈ 0.55 V. While this estimate agrees well with
typical measured values, the vast number of assumptions about
the DOS leads to uncertainties of at least ±0.2 V. It is clear that
more detailed studies of the DOS of donor-acceptor blends are
necessary to make advances in this area.
Our results also suggest that disorder is a possible origin
of the nonideal diode behavior found in most BHJ OPVs. An
intensity-dependent transition from ideal to nonideal behavior
is predicted. One of the notable differences between the GDM
and ME models is that the GDM predicts ideal behavior at low
intensities and nonideal behavior at high intensities, while the
opposite is the case for the ME model. Experimentally, this
transition is not usually observed, though the range of intensities that can be studied can be quite limited due parasitic series
and shunt resistances. It remains difficult to choose between the
models, and some transport studies even combine them both.71
In both models, the nonideality coincides with a dependence
of mobility on carrier density. This can explain observations
of apparent power-law dependence of recombination rate on
intensity,5,17,24–26 though these could also result from voltagedependent nonuniform charge distributions.72 While physically different from Shockley-Read-Hall (SRH) recombination
in homojunction devices, the (f-t) recombination process
treated in this paper can be regarded as a specific class of SRH
recombination with distributions of trap states close to the valence band maximum and conduction band minimum that are
neutral when empty (i.e., nondoping), and that have Langevin
capture cross-sections when occupied. A very recent publication used this approach and arrived independently at the same
nonideal behavior in dark currents.73 Care should be taken not
to overextend this analogy with SRH: It is still unclear how
heterojunction morphology interacts with (f-t) recombination.
Several reports have suggested a strong correlation between
the energy of emissive charge transfer (CT) states and
VOC .7–9,11 It was found that measurements of CT state emission
and absorption energy were more strongly correlated to
VOC than measurements of donor/acceptor oxidation/reduction
potentials. There are two possible explanations. Firstly, the
075210-10
RELATIONSHIP BETWEEN ENERGETIC DISORDER AND . . .
relationship could be coincidental. CT states can be thought of
as hybrid excitons formed between the donor HOMO and acceptor LUMO states. The CT state energy is therefore strongly
related to the donor-acceptor energy gap, and it might be a more
reliable predictor of the effective energy gap than independent
measurements of donor/acceptor valence/conduction state
onsets by other techniques. It is not surprising to find a
correlation between CT state energy and VOC , even if the CT
state plays only a minor role in device performance.
The second explanation is that there is a causal relationship
between CT state energy and VOC . Consider the case that
charge-carrier recombination is mediated through the decay of
radiative CT states. If CT state decay is the primary pathway
to recombination, and CT state decay is slow enough, then
CT states will dissociate into free carriers faster than they can
decay. This will be manifest in a reduced effective recombination rate, which will lead in turn to an increase in VOC . In this
case, the recombination rate will be RCT = nCT /τCT , where
nCT is the density of excited CT states, and τCT is the CT
state lifetime. As charge carriers repeatedly encounter each
other and separate again, the occupancy of CT states will be
in balance with the number of free carriers. Therefore, the
probability of a CT state being occupied will depend on the
applied voltage and the energy of the CT state. Under these
specific conditions, VOC depends not only on the CT state
energy, but also the lifetime, density, and energetic distribution
of CT states. To some extent, the CT state energy is already
included via the Braun treatment of bound-pair dissociation,11
though there is confusion over the definition of a CT state. In
the context of the Braun model, it refers to a Coulombically
bound pair of mobile charge carriers that can have a range of
the order of 10 nm. In the context of spectroscopy, it refers to
a highly localized emissive interfacial excitation.
Vandewal et al.10 showed by detailed balance that efficient
OPVs should also make efficient electroluminescent devices,
*
james.blakesley@physics.org
S. H. Park, A. Roy, S. Beaupré, S. Cho, N. Coates, J. S. Moon, D.
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1
PHYSICAL REVIEW B 84, 075210 (2011)
because suppression of nonradiative recombination will lead
to a higher VOC and quantum efficiency. The highest electroluminescent quantum efficiency of the OPVs they measured was
of the order of 0.0001%, demonstrating that the vast majority
of bimolecular recombination is nonradiative. It is not clear
whether spectroscopy of radiative CT states alone gives much
insight into the dominant recombination pathways. The CTlimited recombination hypothesis also fails to explain so far the
super-bimolecular recombination behavior. While this hypothesis should not be ruled out, further experiments are required.
VI. CONCLUSIONS
Numerical device simulations including field-dependent
bound-pair dissociation, mirror charge potentials, and two
different models of energetic disorder were performed. The
results demonstrated that energetic disorder can be responsible
for a loss of open-circuit voltage in BHJ OPVs. The results
of the numerical simulations can be described by a simple
analytical expression in many, but not all, cases. We found
that energetic disorder can also account for the nonideal diode
behavior and intensity-dependent recombination rates often
observed in real OPVs.
While the link between VOC and energetic disorder is not
yet proven, there is a strong incentive for the community to
improve methods of quantifying the distribution of tail states,
and to study their impact on device performance. Tackling the
effects of energetic disorder could be a key to improving OPV
efficiency beyond 10%.
ACKNOWLEDGMENTS
J.C.B. would like to thank Neil Greenham of the University
of Cambridge for interesting discussions that fomented this
work.
9
K. Tvingstedt, K. Vandewal, A. Gadisa, F. Zhang, J. Manca, and O.
Ingenäs, J. Amer. Chem. Soc. 131, 11819 (2009).
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K. Vandewal, K. Tvingstedt, A. Gadisa, J. V. Manca, Nat. Mater. 8,
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(2010).
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